Problem: $ A = \left[\begin{array}{rrr}-1 & 4 & 5 \\ 2 & 1 & 2\end{array}\right]$ $ C = \left[\begin{array}{rr}-1 & 1 \\ 4 & 1 \\ 0 & -1\end{array}\right]$ What is $ A C$ ?
Solution: Because $ A$ has dimensions $(2\times3)$ and $ C$ has dimensions $(3\times2)$ , the answer matrix will have dimensions $(2\times2)$ $ A C = \left[\begin{array}{rrr}{-1} & {4} & {5} \\ {2} & {1} & {2}\end{array}\right] \left[\begin{array}{rr}{-1} & \color{#DF0030}{1} \\ {4} & \color{#DF0030}{1} \\ {0} & \color{#DF0030}{-1}\end{array}\right] = \left[\begin{array}{rr}? & ? \\ ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ A$ , with the corresponding elements in column $j$ of the second matrix, $ C$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ A$ with the first element in ${\text{column }1}$ of $ C$ , then multiply the second element in ${\text{row }1}$ of $ A$ with the second element in ${\text{column }1}$ of $ C$ , and so on. Add the products together. $ \left[\begin{array}{rr}{-1}\cdot{-1}+{4}\cdot{4}+{5}\cdot{0} & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ A$ with the corresponding elements in ${\text{column }1}$ of $ C$ and add the products together. $ \left[\begin{array}{rr}{-1}\cdot{-1}+{4}\cdot{4}+{5}\cdot{0} & ? \\ {2}\cdot{-1}+{1}\cdot{4}+{2}\cdot{0} & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ A$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ C$ and add the products together. $ \left[\begin{array}{rr}{-1}\cdot{-1}+{4}\cdot{4}+{5}\cdot{0} & {-1}\cdot\color{#DF0030}{1}+{4}\cdot\color{#DF0030}{1}+{5}\cdot\color{#DF0030}{-1} \\ {2}\cdot{-1}+{1}\cdot{4}+{2}\cdot{0} & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rr}{-1}\cdot{-1}+{4}\cdot{4}+{5}\cdot{0} & {-1}\cdot\color{#DF0030}{1}+{4}\cdot\color{#DF0030}{1}+{5}\cdot\color{#DF0030}{-1} \\ {2}\cdot{-1}+{1}\cdot{4}+{2}\cdot{0} & {2}\cdot\color{#DF0030}{1}+{1}\cdot\color{#DF0030}{1}+{2}\cdot\color{#DF0030}{-1}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rr}17 & -2 \\ 2 & 1\end{array}\right] $